Lemma 14 (Theorem B.1 of [1]}) 

Let \(\lbrace X_t\rbrace _{t \ge 1}\)  be Polish spaces and \(\lbrace Y_t\rbrace _{t \ge 1}\)  be countable sets. Consider a Gale-Stewart game whose set \(\mathcal {W}\subseteq \prod _{t \ge 1} (X_t \times Y_t)\)  of winning strategies for \(\texttt {P}_L\)  is finitely decidable and coanalytic. Then there is a universally measurable winning strategy.

